Tuesday, November 11, 2014

The EPR Paradox and Quantum Nonlocality (Part 2)

Continued from Part One:3. Bohm’s Version of Quantum Mechanics

Regarding
the various violations of the Bell Inequality, David Bohmwith colleague Brian Hiley developed an
alternative form of quantum mechanics to integrate it within existing
observations. This began by treating the de Broglie wave as a physically real
entity not merely a statistical one.Also, Bohm and Hiley refined the concept of the pilot wave, originally
proposed by Louis de Broglie

If matter waves, or de Broglie waves, can
be “piloted” then clearly they will have far more theoretical impact than if
mere products of random encounters or observations. Hence, Bohm and Hiley ‘s
development of a “pilot wave” theory to accompany the acceptance of B-waves
physical reality.

To
understand the “clocking” guidance system for these waves, we begin with the basic
energy definition for the quantum given some rest frequency, f o.

Then the energy quantum associated with this frequency is, by Planck’s
equation:

Eo = h fo

Where h is the Planck
constant. Then we can also write:

fo = Eo / h

In the relativistic limit, for photons: fo = mo c2 / hNow, change to angular frequency wo to make the
synchronous mechanism consistent with that proposed by de Broglie and Bohm,
Hiley[1]. Then:

There is also an additional condition, known as the Bohr-Sommerfield condition,
for the clock to remain in phase with the pilot wave:

∮ p dx = n ħ

Now, the momentum p = mo
c , so that the integral becomes:

2p x (mo c) = n ħ

And
the de Broglie wavelength emergence (lD = h/ p) is evident in
the equation. In this sense, we have:

2p x = n
(h/ mo c)
= n lD

Or
the same expression (2p
r = n lD) for the standing waves in an
atom. In
Bohm’s own development, the procession of B-waves is actually enfolded within a
“packet” of P-waves[2]. A basic diagram of the arrangement is shown
below.

The
axis labeled E is actually the real part of the electric field component, Ez.
The width of the p-wave packet is denoted by the spread:

Dk = p / (x – xo)

Where
xo denotes the center point
of the wave packet. In other words, if the center point is at xo = 0, the packet width is just Dk = p / x.

The
wavelength, l = 2p / Dk then is much
less than the width of the packet. E.g. if Dk = p / x then:

l = 2p / (p / x) = 2x. so if x = 1
nm, then l = 2 nm and:

Dk = p / x = p / (1 nm) = p nm, but p (nm) > 2 nm.

The maximum of the wave packet is
approximated closely by the square of the amplitude:

[
Ez ] 2 = 4 sin2Dk (x – xo) /
(x – xo) 2

We can check the limits of the preceding.
Let xo = 0 then:

[
Ez ] 2 = 4 sin2Dk x / x 2

And:

[
Ez ] 2 = 4 sin2Dk / x

Conversely,
let xo = x, then:

[ Ez ] 2 = 0 = 4 sin2Dk (x – x) / (x – x) 2

Thus,
the p-wave packet ceases to exist as a discrete or localized entity and thereby
loses its particle properties. But
what about the photon mass?

Mass
can be derived as a basis for the wave packet spread Dx = (x – xo). Thus, a “particle” is
represented as a finite wave packet with wavelength-based spread Dl, such that, according
to the Heisenberg Uncertainty Principle:

Given
a spread in the velocity D
v = D k =
D (vg) then the Heisenberg
Uncertainty Principle states:

D x D p » ħ
where D
p =
D (m vg)

Therefore:
D x D (m vg) » ħ

Or,
since the “ultimate” lower limit on D x is of the order of the Compton wavelength,
i.e. D x
»x or ħ/
m c, we have:

m
» ħ/ D x c

Where
D x denotes a lower limit to D x. And it can be shown
that the quantity 1/ x
possesses the additive and inertial properties of mass.

4. Bohm’s Version of the Uncertainty Principle:

In
Bohm’s case, the Heisenberg relations are embodied in his theory as a limiting
case over a certain level of intervals of space and time. However, the
potential exists for the fields to be averaged over smaller intervals and
hence, subject to a greater degree of self-determination than is consistent
with the Heisenberg principle. As Bohm concludes[3]:

From this,
it follows that our new theory is able to reproduce, in essence at least, one
of the essential features of the quantum theory, i.e. Heisenberg’s principle,
and yet have a different content in new levels

Bohm
is primarily concerned with the canonically conjugate field momentum, for which
the associated coordinates, i.e. Dt,
Dfk fluctuate at random. Thus, we have, according
to Bohm:

pk = a (Dfk /Dt)

Where a is a constant of proportionality, and Dfk is the fluctuation of the field coordinate.
If then the field fluctuates in a random way the region over which it
fluctuates is;

(dDfk)2 = b (Dt)

Taking
the square root of both sides yields:

(dDfk) = b 1/2 (Dt)1/2

Bohm
notes that pk also fluctuates at random
over the given range so:

dpk = a b 1/2 / (Dt)1/2

Combining
all the preceding results one finally gets a relation reflective of the
Heisenberg principle, but time independent:

dpk (dDfk ) = ab

This
is analogous to Heisenberg’s principle, cf.

dp dq <
ħ

Where
the product ab plays the same
role as ħ

Problems:

1) Consider the details of the Einstein ‘box’
thought experiment in which he attempted to out wit Bohr. Based on the
hypothetic measured quantities given, how much uncertainty would be expected in
the mass? In the weight? (Find the mass uncertainty in kg, and the weight in
N.)

2)An
Aspect-style experiment is carried out and th e
result is found to be:

4) In a particular experiment to test Bohmian
quantum mechanics on a computer, the uncertainty in one input turns out to be: Dt = 10 -39
s and in the other, Dfk = 10 -51 m. From this data, find
the quantity b. Then compose a form of the Uncertainty principle and obtain the
product ab. (Where ab plays the same role as
ħ in the conventional form of the Uncertainty principle.)

About Me

Specialized in space physics and solar physics, developed first astronomy curriculum for Caribbean secondary schools, has written thirteen books - the most recent:Fundamentals of Solar Physics. Also: Modern Physics: Notes, Problems and Solutions;:'Beyond Atheism, Beyond God', Astronomy & Astrophysics: Notes, Problems and Solutions', 'Physics Notes for Advanced Level&#39, Mathematical Excursions in Brane Space, Selected Analyses in Solar Flare Plasma Dynamics; and 'A History of Caribbean Secondary School Astronomy'. It details the background to my development and implementation of the first ever astronomy curriculum for secondary schools in the Caribbean.